The simulation results are illustrated as in Figure REF , with the ambient PSD cone of size \(n = 20\) . Here, the red curve \(SDD-190\)  denotes the case when \(\mathcal {C} = SDD_n\) , and 190 indicates the number of sub-cones \(\binom{20}{2}=190\) .
Similarly, the dark blue curve \(FW3-1140\)  represents the approximations using \(\binom{20}{3}=1140\)  factor-width 3 matrices \(\mathcal {C} = \mathcal {FW}_n^3\) .
The other three curves named by Frame-N denote a family of inner approximations which takes the form \( \mathcal {C}= \sum _{k=1}^N F_k[ \mathbb {S}_+^s]F_k^T\)  as in (REF ).
Here, four different packing have been found using the alternating projection algorithm [1]}, where the numbers 1,30,190,350 represents the number N of sub-cones (equivalently number of frames). In each case, we gradually increase the sub-cone dimension \(s\)  to obtain tighter inner approximations of ambient PSD cone \(\mathbb {S}^n_+\) .
The mean distance is calculated as the mean of 100 random tests. A more accurate inner approximation is then indicated by smaller mean distance. Notably, when the sub-cones are spanned by \(\mathbb {S}_+^{10}\) , one could reach almost perfect approximations with only \( N = 30\)  frames, compared with 190 sub-cones for SDD approach and 1140 sub-cones for FW3 approach. Also, instead of using 1140 sub-cones as in FW3 approach, we achieve a similar approximation accuracy with only 350 sub-cones.
